GATE EC 2026 Set 1 Q10 — Combinatorial Circuits

A Boolean function, $f(x, y, z)$ with $x$ as MSB and $z$ as LSB is realized by $4: 1$ multiplexer (MUX) with select lines, $\mathrm{S}_{1}$ and $\mathrm{S}_{0}\left(\mathrm{~S}_{1}\right.$ is MSB, $\mathrm{S}_{0}$ is LSB) and inputs, $\mathrm{I}_{0}, \mathrm{I}_{1}, \mathrm{I}_{2}, \mathrm{I}_{3}$ as shown in the Figure. Which of the …

Consider the four-variable Boolean function, $f(w, x, y, z)=\sum m(0,2,5,7,8,10,13,14,15)$ with '$w$' as MSB and '$z$' as LSB. Which of the following expressions is/are the valid form(s) of $f(w, x, y, z)?$

A $3$-input majority logic gate has inputs $X, Y$, and $Z$. The output $F$ of the gate is logic '$1$' if two or more of the inputs are logic '$1$'. The output $F$ is logic '$0$' if two or more of the inputs are logic '$0$'. Which one of the following options is a Boolean expression of the output $F$ ?

A full adder and an XOR gate are used to design a digital circuit with inputs $X, Y$, and $Z$, and output $F$, as shown below. The input $Z$ is connected to the carry-in input of the full adder. If the input $Z$ is set to logic ' $l$ ', then the circuit functions as $\_\_\_\_\_\_\_$ with $X$ and $Y$ as inputs.

​​​​For the Boolean function $$F(A, B, C, D)=\sum m(0,2,5,7,8,10,12,13,14,15) \text{, }$$ the essential prime implicants are $\_\_\_\_\_\_$.

As shown in the circuit, the initial voltage across the capacitor is $10 \mathrm{~V}$, with the switch being open. The switch is then closed at $t=0$. The total energy dissipated in the ideal Zener diode $\left(V_{Z}=5 \mathrm{~V}\right)$ after the switch is closed (in $\mathrm{mJ}$, rounded off to three decimal places) is $\_\_\_\_\_\_$.